Fixed-Income Forward and Futures Contracts

Mitchell Global Group wants to enter into a fixed-income forward contract. As you can imagine, with the bond market's enormous values, forward contracts are plentiful, but the terms aren't always the same. What's one reason that might be the case?

No. Countries will recognize the same par value for the same bond.

No. Countries will recognize the same interest rate for the same bond.

Yes! Accrued interest will impact the price of the same bond because some countries include the accrued interest in the price (dirty price), while other countries quote the bond price without accrued interest (clean price). So Mitchell would first need to determine which country the forward rate bond will be priced in. Accrued interest (AI) is calculated based on the number of accrued days (NAD), number of total days in the coupon (NTD), the coupon itself (C), and periodicity (n). So for example, a 2% coupon on a USD 1,000 par bond coupon 43 days after the last semiannual coupon would suggest accrued interest of: $$\displaystyle AI = \frac{NAD}{NTD} \times \frac{C}{n} = \frac{43}{180} \times \frac{20}{2} \approx 2.39 $$

But that's not all that Mitchell needs to consider. Consider the fact that fixed-income forward contracts allow the seller to deliver a range of bonds, not just one specific issuance. So fixed-income forward contracts include a conversion factor that's applied to compute the forward contract price. What's the purpose of the factor?

You got it! The conversion factor is a mathematical adjustment to the amount required when settling a futures contract that is supposed to make all eligible bonds equal the same amount. And that typically leads to one bond being identified as the cheapest to deliver.

No. Typically, the cheapest-to-deliver bond will be chosen for delivery after the conversion factor is applied, but that's not the purpose of the factor.

No. If a range of bonds are eligible for delivery, then the most expense bond won't be chosen, so that's not the purpose of the factor.

In Mitchell's case, pricing the fixed-income forward ($$F$$) contract has a few different factors, but thankfully, the overall pricing equation is similar to other assets that have cash flows during the forward contract period. To find the price of the fixed-income forward or futures price including the conversion factor ($$CF$$), use the equation $$\displaystyle F_0 = Q_0CF = FV[S_0 + CC_0 - CB_0] = FV[B_0 + AI_0 - PVCI]$$ which includes the quoted futures price ($$Q_0$$), the spot price ($$S_0$$), the present value of carrying costs ($$CC_0$$), the present value of carry benefits ($$CB_0$$), The quoted bond price ($$B_0$$), accrued interest ($$AI_0$$), and the present value of all coupon interest ($$PVCI$$).

Think for a minute about this equation and its structure in relation to a forward contract with cash flows. $$B_0$$ is the bond price and $$AI_0$$ is the current amount of any accrued interest that's not included in the quoted price. What would $$PVCI$$ have to represent?

No. It's not the future value of all interest. Mitchell is trying the find the forward contract price, not its value.

No. Carry costs would be added to the equation, not subtracted.

Exactly! The bond has cash flows that should be subtracted from the equation because it's a carry benefit. So the breakdown of the equation is the quoted bond price over the current time to maturity plus any accrued interest not included in the price, less the present value of all coupon interest paid over the forward contract horizon.

So for Mitchell, after the forward contract is priced, then the setup of the transaction is similar to other forward contracts. First, Mitchell would buy the bond with borrowed funds, then sell the forward contract and borrow any arbitrage profit. In equilibrium, to eliminate arbitrage opportunities, Mitchell would expect $$\displaystyle F_0 + AI_T = FV[B_0 + AI_0 - PVCI] $$ which also means $$\displaystyle F_0 = FV(S_0) - AI_T - FVCI $$ And substitute the spot price with the forward price to solve for the quoted futures price: $$\displaystyle Q_0 = \left[ \frac{1}{CF} \right] \left( FV[B_0 + AI_0] - AI_T - FVCI \right) $$

To summarize: [[summary]]

But how would Mitchell price a fixed-income futures contract? Well, those contracts are priced similarly to other futures contracts. So what do you think the contract value would be at the end of the day?

Way to go! The futures value will be zero because of the daily settlement. So there's no value left at the end of each day. So fixed-income futures are priced just like other futures: due to the mark-to-market settlement, a bond future's value is just the price change since the previous day, and it is then closed to zero at the end of each day with settlement.

No. Futures are settled each day, so the bond's value doesn't impact the futures value.

No, actually. The price change is settled each day.

Different countries quote the same bonds at different prices due to par values

Different countries quote the same bonds at different prices due to interest rates

Different countries quote the same bonds at different prices due to accrued interest

To find the cheapest-to-deliver bond

To find the most expense bond to deliver

To equalize all eligible bonds for delivery

The future value of all interest paid

The carry costs associated with the bond

The present value of the bond's fixed coupon payments over the forward contract period

Zero

The bond's current quoted price

Price change since previous day

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